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Home TOPICS CDF Tools for Heat Transfer 1D Transient Heat Conduction CDF Tool

1D Transient Heat Conduction CDF Tool

Analytic Solution for 1D Transient Heat Conduction

The problem geometry and boundary conditions are shown below. An initially isothermal (Tinitial) semi-infinite medium is suddenly subject to a surface temperature Th.


The temperature field can be non-dimensionalized as:

\theta (x,t)=\frac{T(x,t)-T_{\text{initial}}}{T_h-T_{\text{initial}}}

The governing differential equation (with spatially one-dimensional heat flow) is

\frac{\partial \theta (x,t)}{\partial t} = \alpha \frac{\partial^2 \theta (x,t)}{\partial x^2}

The solution for all locations x and times t is:

\theta(x,t) = 1-\text{erf}\left[\frac{x}{2\sqrt{\alpha  t}}\right]

where α is the material’s thermal diffusivity.

Graphical CDF Tool

The following is an embedded, active Mathematica CDF tool. The units for α are cm2/sec, with corresponding units of cm and sec for x and t, respectively.

Created on , Last modified on, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.